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selfAdjoint

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## Main Question or Discussion Point

In the QM forum, Vanesch posted this:

Back in the 1950s Fermi, Pasta, and Ulam devised a model, intended to resemble a linear molecule, which consisted of anharmonic oscillators joined by weakly nonlinear couplings. The model was beyond their analytical capabilities but they applied their brand-new monte carlo computer simulation to it and found a surprise. They had expected the model (later named the FPU model after them) to show ergodic behavior at late times, with the different energy states smearing out to fill the phase space. But what the computer output showed them was a nondecreasing propensity to produce unsmeared energy spikes.

It was later shown, still by computer, that the FPU model was producing solitons. Still later the FPU equation was mapped into the discretized Kortweg-deVrees (KdV) equation; the KdV equation has a rich analytical tradition,and its general solution can be expressed as a sum of solitons.

Finally it was shown that if you allow the FPU oscillators to collide and rebound, then the model behaves ergodically. So perhaps elastic collisions are a prior requirement for ergodicity? Does anybody know?

I know a little about the modern history of the ergodic hypothesis and here it is.A probability distribution has always been, in classical physics, an ignorance description, and the funny thing about ergodicity is that the very same mechanisms which make us ignore in practice how DIFFERENT SYSTEMS are prepared, also have as a consequence that different microparts are distributed in time and between them according to the same distributions. I think that this hypothesis has indeed not been proven in all generality, but is assumed and is at the basis of about all thermodynamical calculations.

Back in the 1950s Fermi, Pasta, and Ulam devised a model, intended to resemble a linear molecule, which consisted of anharmonic oscillators joined by weakly nonlinear couplings. The model was beyond their analytical capabilities but they applied their brand-new monte carlo computer simulation to it and found a surprise. They had expected the model (later named the FPU model after them) to show ergodic behavior at late times, with the different energy states smearing out to fill the phase space. But what the computer output showed them was a nondecreasing propensity to produce unsmeared energy spikes.

It was later shown, still by computer, that the FPU model was producing solitons. Still later the FPU equation was mapped into the discretized Kortweg-deVrees (KdV) equation; the KdV equation has a rich analytical tradition,and its general solution can be expressed as a sum of solitons.

Finally it was shown that if you allow the FPU oscillators to collide and rebound, then the model behaves ergodically. So perhaps elastic collisions are a prior requirement for ergodicity? Does anybody know?